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Mon. Not. R. Astron. Soc. 000, 000–000 (0000) Printed 2 September 2016 (MN LATEX style file v2.2)

Variability, polarimetry, and timing properties of single

pulses from PSR J1713+0747 using the Large European

Array for Pulsars

K. Liu,1,2⋆ C. G. Bassa,3 G. H. Janssen,3 R. Karuppusamy,1 J. McKee,4 M. Kramer,1,4

K. J. Lee,5 D. Perrodin,6 M. Purver,4 S. Sanidas,7,3 R. Smits,3 B. W. Stappers,4

P. Weltevrede4 and W. W. Zhu,11Max-Planck-Institut fur Radioastronomie, Auf dem Hugel 69, D-53121 Bonn, Germany2Station de radioastronomie de Nancay, Observatoire de Paris, CNRS/INSU, F-18330 Nancay, France3ASTRON, the Netherlands Institute for Radio Astronomy, Postbus 2, NL-7990 AA, Dwingeloo, The Netherlands4University of Manchester, Jodrell Bank Centre for Astrophysics, Alan Turing Building, Manchester M13 9PL, UK5KIAA, Peking University, Beijing 100871, P.R. China6INAF - Osservatorio Astronomico di Cagliari, Via della Scienza 5, I-09047 Selargius (CA), Italy7Anton Pannekoek Institute for Astronomy, University of Amsterdam, Science Park 904, NL-1098 XH Amsterdam, The Netherlands

2 September 2016

ABSTRACT

Single pulses preserve information about the pulsar radio emission and propagation inthe pulsar magnetosphere, and understanding the behaviour of their variability is es-sential for estimating the fundamental limit on the achievable pulsar timing precision.Here we report the findings of our analysis of single pulses from PSR J1713+0747 withdata collected by the Large European Array for Pulsars (LEAP). We present statisti-cal studies of the pulse properties that include distributions of their energy, phase andwidth. Two modes of systematic sub-pulse drifting have been detected, with a peri-odicity of 7 and 3 pulse periods. The two modes appear at different ranges of pulselongitude but overlap under the main peak of the integrated profile. No evidence forpulse micro-structure is seen with a time resolution down to 140 ns. In addition, weshow that the fractional polarisation of single pulses increases with their pulse peakflux density. By mapping the probability density of linear polarisation position anglewith pulse longitude, we reveal the existence of two orthogonal polarisation modes.Finally, we find that the resulting phase jitter of integrated profiles caused by singlepulse variability can be described by a Gaussian probability distribution only when atleast 100 pulses are used for integration. Pulses of different flux densities and widthscontribute approximately equally to the phase jitter, and no improvement on timingprecision is achieved by using a sub-set of pulses with a specific range of flux densityor width.

Key words: methods: data analysis — pulsars: individual (PSR J1713+0747)

1 INTRODUCTION

Millisecond pulsars (MSPs) that were spun up in accret-ing binary systems to reach rotational periods . 30 ms(Alpar et al. 1982), are noted for their highly precise timingbehaviour (e.g. Arzoumanian et al. 2015; Desvignes et al.2016; Reardon et al. 2016). Their short and stable rota-tional period make them excellent tools for probing tiny

⋆ kliu@mpifr-bonn.mpg.de

spacetime perturbations and performing gravity experi-ments, including tests of General Relativity with greatprecision (e.g. Kramer et al. 2006; Weisberg et al. 2010),stringent constraints on alternative theories of gravity (e.g.Freire et al. 2012; Antoniadis et al. 2013), probes of neutronstar equations-of-state (e.g. Demorest et al. 2010; Ozel et al.2010), and the ongoing search for gravitational waves in thenanohertz regime (e.g. Lentati et al. 2015; Shannon et al.2015; Arzoumanian et al. 2016).

The success of the aforementioned timing experimentsis attributed to both the regular rotation of the pulsars,

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2 K. Liu et al.

and their stable integrated pulse profiles formed by averag-ing over tens of thousands of periods. Nevertheless, it hasbeen known since the discovery of pulsars that the pulsedemission from every single rotation (thus single pulses) ofa pulsar is highly variable. This was first noticed in canon-ical pulsars and more recently in MSPs (e.g. Jenet et al.1998; Shannon & Cordes 2012; Liu et al. 2015). A conse-quence of such variability is the so-called pulse phase jitterphenomenon in integrated profiles, which for a given inte-gration time places a fundamental limit on the achievabletiming precision on short timescales. Timing precision hasalready reached the jitter-limited regime for a few MSPs(Liu et al. 2012; Shannon & Cordes 2012), and this limitis expected to be reached for many more when the nextgeneration of radio telescopes (e.g., the Square KilometreArray) comes online (Liu et al. 2011; Janssen et al. 2015).Most recent timing analysis has started to take into accountthe effect of phase jitter when modelling the noise in thetiming data (e.g. Lentati & Shannon 2015; Zhu et al. 2015;Caballero et al. 2016). Detailed studies of single pulse vari-ability in MSPs are crucial for building a comprehensiveunderstanding of this phenomenon, if we are to push be-yond this limitation. Such investigations will also provideinput for the efforts to either model or mitigate jitter noisein timing data (Os lowski et al. 2011; Imgrund et al. 2015).

The origin of pulsar radio emission is associated withplasma processes in the highly-magnetised pulsar mag-netosphere (e.g. Cheng & Ruderman 1977; Cordes 1979;Cairns et al. 2004), the understanding of which has still beenelusive. Single pulse data preserve information of the inten-sity, polarisation, and even waveform (if dual-polarizationNyquist sampled time series are recorded) of the emis-sion from every single rotation. Studying single pulses canshed light on the nature of the pulsar emission mechanism,by, e.g., revealing the fundamental units of coherent ra-diation (Cordes 1976; Gil 1985; Jenet et al. 2001), distin-guishing different modes of polarised emission (Gil & Lyne1995; Edwards & Stappers 2004), characterising the tempo-ral variability in pulse intensity (Ruderman & Sutherland1975; Gil et al. 2003; Weltevrede et al. 2006), and so forth.Studying pulsars that display the drifting sub-pulse phe-nomenon can also assist in determining the viewing geom-etry and provide insight into the structure of the emissionregion (Drake & Craft 1968; Ruderman & Sutherland 1975;Rankin 1986; Qiao et al. 2004). The vast majority of singlepulse emission studies has been mostly of canonical (normal,non-recycled) pulsars (see Lyne & Graham-Smith 2006, foran overview). Extending this work to include MSPs willestablish a bridge between the understanding of emissionphysics in canonical pulsars and that in MSPs, so as to seeif a common theory or model of pulsar radio emission canbe applied.

A small fraction of MSPs emit occasional giant radiopulses, which have been studied in detail (Cognard et al.1996; Knight et al. 2006; Knight 2007; Zhuravlev et al.2013; Bilous et al. 2015). However, investigations into or-dinary single pulses of MSPs have been carried out onlyfor a limited number of bright sources (Jenet et al. 1998;Edwards & Stappers 2003; Shannon & Cordes 2012; Bilous2012; Os lowski et al. 2014; Shannon et al. 2014; Liu et al.2015), one of which is PSR J1713+0747. This pulsar isone of the most precisely timed pulsars and has been

included in current pulsar timing array campaigns todetect gravitational waves in the nanohertz frequencyrange (Verbiest et al. 2016). The binary system it inhab-its is also an ideal test laboratory for alternative theo-ries of gravity (Zhu et al. 2015). Edwards & Stappers (2003)showed that there is clear modulation in the pulses fromPSR J1713+0747 and that it varies across the pulse pro-file and as a function of observing frequency. In the fluctu-ation spectra, they showed that the pulsar exhibited twobroad maxima corresponding to fluctuations at 0.17 and0.35 cycles-per-period (cpp). However, the longitude depen-dence and a correlation with drifting sub-pulses could not beestablished due to the lack of sensitivity. Shannon & Cordes(2012) have shown that single pulses from PSR J1713+0747are highly variable in phase, which already limits its timingprecision with the current observing sensitivity. Thus, un-derstanding single pulse variability of this pulsar in order topotentially mitigate its contribution to the timing noise isclearly necessary and will be the focus of this work.

The rest of this paper is structured as follows. In Sec-tion 2, we provide the details of the observation and pre-processing of the data. Section 3 presents the results fromour data analysis on single pulse variability, polarisation andtiming properties. We conclude in Section 4 with a brief dis-cussion and prospects for future work.

2 OBSERVATIONS

Investigations of single pulses from MSPs have been greatlyrestricted by their comparatively low flux density in the ra-dio band, as well as the lack of availability of single pulsedata. This is because of the large data volumes and precisiontiming typically being performed on integrated pulse data.The Large European Array for Pulsars (LEAP) performsmonthly simultaneous observations at 1.4 GHz of twenty-three MSPs with the five 100-m class radio telescopes inEurope (Bassa et al. 2016). Coherently combining the volt-age data from the radio telescopes delivers the sensitivityequivalent to a single dish with diameter up to 195 m andgain up to 5.7 K Jy−1. For all pulsars observed, the productof coherently-added voltages can be used to generate sin-gle pulse data with full polarisation information and vari-able time resolution. In addition to LEAP’s high sensitivity,the monthly observing campaign increases the chances ofachieving a high signal-to-noise (S/N) detection, especiallyfor low-DM pulsars, whose flux density varies dramaticallydue to diffractive interstellar scintillation. Therefore, LEAPprovides a database which is ideal for single pulse studies ofMSPs.

For the study in this paper, we selected 15 minutes ofarchived data obtained on MJD 56193 (23 September 2012)when a bright scintle was caught in the LEAP frequencyband. The observation was conducted simultaneously withthe Effelsberg Radio Telescope, the Nancay Radio telescope,and the Westerbork Synthesis Radio Telescope (WSRT)1.At Effelsberg, the PSRIX pulsar backend was used in base-band mode to record 8×16 MHz bands of two orthogo-

1 The Lovell Telescope at Jodrell Bank was under maintenanceduring the observation.

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Single pulses of PSR J1713+0747 3

nal polarisations at the Nyquist rate and with 8-bit sam-pling (Lazarus et al. 2016). The eight bands are centred at1340, 1356, 1372, 1388, 1404, 1420, 1436, and 1452 MHz. AtNancay, we used the NUPPI instrument (Desvignes et al.2011) to record baseband data from four of the eight bands(1388, 1404, 1420, 1436 MHz) with an identical setup. At theWSRT, the PuMa-II system (Karuppusamy et al. 2008) wasused to record 8-bit baseband data of 8×20 MHz bands, withcentral frequencies at 1342, 1358, 1374, 1390, 1406, 1422,1438 and 1454 MHz2. At both Effelsberg and the WSRT, thedata were later copied to disks and shipped to the JodrellBank Observatory (JBO), where they were installed into theLEAP central storage cluster. The data from Nancay weredirectly transferred to JBO via internet. A full description ofthe LEAP observational setup can be found in Bassa et al.(2016).

Preprocessing of the data was then carried out on theCPU cluster at JBO using a software correlator developedspecifically for LEAP (Smits et al. in prep.). Each obser-vation was divided into 3-min sub-integrations for corre-lation purposes. To calibrate the polarisation of the volt-age data from each individual telescope, we first dedis-persed and folded the data to form an integrated profilefrom the entire observation. Using the polarisation profileof PSR J1713+0747 (from Stairs et al. 1999) obtained fromthe European Pulsar Network (EPN) database3, we applieda template matching technique to the observed profile inorder to measure the receiver properties, assuming the fullreception model as in van Straten (2004). The correspondingJones matrices were then calculated and used to calibratethe voltage data. In order to correct for the difference in volt-age phase response between different backends, we used theobservation of the phase calibrator J1719+0817 (performedimmediately before the pulsar observation) to measure thephase offsets across the entire band, and then applied themto the data of PSR J1713+0747. As it was a very brightobservation of the pulsar, we used the pulsar data itself tomeasure the fringes (relative time and phase offsets) betweendifferent sites. With the fringe solutions, the software cor-relator produced coherently added voltage data with an ef-fective bandwidth of 112 MHz centred at 1388 MHz4, whichwere then processed with the DSPSR software (for details,see van Straten & Bailes 2011) to extract the single pulsedata. The data were coherently dedispersed using a disper-sion measure (DM) of 15.9891 cm−3pc (which is the defaultvalue used at JBO to de-disperse PSR J1713+0747 data).Single pulse data were written to disks with a maximumtime resolution of 140 ns (corresponding to 32,768 phase binswithin a period). More details of our data processing pipelineare described in Bassa et al. (2016).

Fig. 1 shows the total intensity profile from the LEAPdata compared with those from each of the single telescopes.A calculation based on signal-to-noise ratio measurementsshows a coherency of 95% for the LEAP addition. The polar-isation profile as well as the swing of linear polarisation posi-tion angle (P.A.) from the LEAP data shown in Fig. 2 is con-

2 The WSRT bands overlap by 4MHz, so the coherent additionlater used 16MHz out of the 20MHz.3 http://www.epta.eu.org/epndb/4 Data from the 1452MHz sub-band were not included due to areceiver cutoff at Effelsberg.

0

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NCY

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Figure 1. Total intensity profile of PSR J1713+0747 on

MJD 56193 from LEAP data integrated over the entire 15-minobservation, compared with those from single telescope data. Theamplitude of the profiles from individual telescopes are relativeto that of the LEAP profile. The peak signal-to-noise ratios are896, 314, 333, 300 for LEAP, Effelsberg (EFF), Nancay (NCY),and Westerbork (WSRT), respectively, which corresponds to anaddition with a 95% coherency. The peak of the LEAP profilelocates at phase 0.507.

Figure 2. Polarisation profile and linear polarisation positionangle (P.A.) of PSR J1713+0747. In the lower panel, the solid(black), dashed (red) and dotted (blue) lines represent the totalintensity (I), linear (L) and circular (V ) components, respectively.The inner panel shows an enlarged view of the polarisation profile.The time resolution used is 2.3µs.

sistent with the results from previous work (Ord et al. 2004;Yan et al. 2011). We also used the receiver properties mea-sured from PSR J1713+0747 to calibrate PSR J1600−3053data obtained on the same epoch, and had consistent resultas shown in the EPN database.

The variability of single pulse emission is commonly in-vestigated using a set of statistical tools. The modulationindex is used to indicate the level of intensity variation.The fluctuation (power) spectra of pulse intensity as a func-tion of pulse number and rotational phase (pulse stack),can identify periodicities in the emission. The longitude-resolved fluctuation spectrum (LRFS) resolves the fluctua-tions as a function of rotational phase. The two-dimensionalfluctuation spectrum (2DFS) is calculated by performinga two-dimensional Fourier Transform of the pulse stack,

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4 K. Liu et al.

Figure 3. An example of a pulse stack containing one hundredpulses. The time resolution used here is 2.3µs. The figure il-lustrates the pulse-to-pulse variation which is analysed quanti-tatively further below (see Fig. 7).

which is widely used to study the systematic drifting of sub-pulses. Details of these statistical tools can be found in e.g.,Edwards & Stappers (2003) and Weltevrede et al. (2006).The software tools used to conduct single pulse analysis inthis paper are part of the PSRSALSA project (Weltevrede2016), and are freely available online5.

3 RESULTS

The observation recorded a total of 196,915 single pulses, ap-proximately 75% of which were detected with a peak signal-to-noise ratio (S/N) higher than 5 based on a time resolu-tion of 9µs. The highest peak S/N is 36, a factor of threegreater than the maximum recorded in Shannon & Cordes(2012). Fig. 3 shows an example stack of one hundred con-secutive pulses, which shows clear intensity variation amongthe pulses. During the 15-min observation, the flux densityof the pulsar dropped by approximately 10% due to inter-stellar scintillation.

3.1 Single pulse properties

The energy distribution of single pulses fromPSR J1713+0747 has been studied in previous work(Shannon & Cordes 2012; Shannon et al. 2014), based oneither the peak flux density or the total flux density from achosen on-pulse region. Here we used the single pulse datato study the energy distribution in three phase ranges thatcorrespond to the leading edge, the peak, and the trailingedge of the integrated profile (see Fig. 4). The observedpulse energy was calculated by summing the intensitieswithin the corresponding phase range defined in Table 1after subtracting the off-pulse mean. The correspondingnoise energy distributions were obtained from an equallylong phase range in the off-pulse region. To compensatefor the variation in flux density caused by scintillation, thedata were first re-scaled to ensure that the average pulseenergy in subsequent 30-s intervals remains constant. The

5 https://github.com/weltevrede/psrsalsa

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Figure 4. Pulse energy distributions in phase ranges (detailedin Table 1) corresponding to the leading edge, the peak, andthe trailing edge of the integrated profile. The dashed, dotted,solid, and dashed-dotted lines represent the observed pulse en-ergy distribution, the noise energy distribution, the best-fit tothe observed distribution, and the modelled intrinsic pulse en-ergy distribution with a log-normal function, respectively.

observed energy distribution was modelled by convolving anintrinsic distribution, here assumed to be log-normal, withthe observed noise distribution via a procedure detailedin Weltevrede et al. (2006) and Weltevrede (2016). Thelog-normal probability density function is defined as

ρ(x) =1√

2πxσexp

[

− (lnx− µ)2

2σ2

]

. (1)

For each region, the observed and the modelled intrinsicpulse energy distributions are shown in Fig. 4. It can beseen that the three regions have a different intrinsic pulseenergy distribution, that of the peak region being broadest.A Kolmogolov-Smirnov (K-S) test (summarised in Table 1)shows that in all three cases the intrinsic pulse energy canbe well described by an intrinsically log-normal distribution.

In Fig. 5, we plot the number density distribution ofpulses with respect to their relative peak flux densities andphases of the pulse peak. We note that pulses with higherpeak flux densities tend to be located within a narrow phaserange that includes the maximum of the integrated profile.Lower-amplitude pulses can occur in a wider phase rangewhich extends more to the trailing edge of the integratedprofile. This confirms the finding in Shannon & Cordes(2012), and indicates that the phase distribution of these

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Single pulses of PSR J1713+0747 5

Table 1. Phase (see Fig. 1 to correspond to the integrated profile)ranges defined for the leading edge, the peak, and the trailing edgeof the integrated profile, p-values from a K-S test on the modelledand the observed pulse energy distributions in these three regions,and the best-fit modelling parameters as defined in Eq. 1. Noneof these p-values indicates a significant difference between themodelled and observed distributions.

Phase range p-value µ σ

Leading (0.400, 0.500) 0.99 -0.090 0.37Peak (0.500, 0.515) 0.60 -0.12 0.52

Trailing (0.515, 0.615) 0.67 -0.052 0.30

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Figure 5. Number density distribution (lower panel) of pulseswith respect to their relative peak flux densities and phase ofpulse peak, compared with the integrated profile (upper panel).Here we used a time resolution of 9µs for the data.

pulses is not symmetric. More investigations into the im-pact on timing follow in Section 3.3. The occurrence rateof bright pulses whose relative peak flux densities are largerthan 3 was found to be constant during the observation.

It was shown in Liu et al. (2015) that the bright singlepulses from the MSP J1022+1001 have a preferred pulsewidth for any given peak flux density. To investigate whethera similar feature is present in the pulses of PSR J1713+0747,in Fig. 6 we plotted the number density distribution of pulseswith respect to their relative peak flux density and width.Here we define pulse width as the full width at 50% of thepeak amplitude (i.e., W50), as in e.g., Lyne et al. (2010). Wefound that a pulse width of roughly 0.04 ms is favoured forthe bright single pulses, which is smaller than the W50 ofthe integrated profile (0.11 ms).

To investigate the variability of single pulses, we havecalculated the longitude-resolved modulation index, theLRFS, and the 2DFS based on the entire length of the ob-servation (Fig. 7). Here we used the same definitions as inWeltevrede et al. (2006). The modulation index as a func-tion of pulse phase was found to be consistent with the re-sults in Edwards & Stappers (2003) and Shannon & Cordes(2012), and yields a significantly improved resolution. Itis maximised at the extreme edges of the integrated pro-file, which is commonly seen in canonical pulsars (e.g.Weltevrede et al. 2006). The asymmetric distribution of the

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Figure 6. Number density distribution of pulses with respectto their relative peak flux density and width. The peak bins wereselected based on a time resolution of 9µs. The dashed line showsthe pulse width corresponding to the maximum number densityfor a given peak flux density.

modulation index also indicates that the intensity variationis different in the leading and trailing edges of the inte-grated profile, as expected from the results in Fig. 4. Inthe LRFS shown in Fig. 7, two maxima are visible at 0.14and 0.34 cpp. This detection confirms the discovery of pe-riodic intensity modulation by Edwards & Stappers (2003)with much higher significance. For better visualisation of thetwo modes of periodicity, we subtracted the mean power ineach spectrum (i.e. column) and plotted the modified LRFSin Fig. 8. It can be seen that both periodicities occur at thephase of the main peak of the integrated profile. The modewith periodicity of 0.14 cpp extends further to the trailingedge of the integrated profile, where no significant powerwas detected corresponding to the periodicity of 0.34 cpp.While similar phenomena are known to occur in canoni-cal pulsars (e.g. Kramer et al. 2002), this is the first time aphase-dependent periodicity of intensity variations has beenreported for a MSP.

In the 2DFS, the two local maxima at 0.14 and 0.34 cppare clearly offset from the vertical axis of zero, indicating theassociation of the two periodicities with systematic driftingof emission power in pulse phase. Following the descriptionin Weltevrede et al. (2006), we have measured the horizon-tal separation of the drifting in pulse longitude (P2) andtheir vertical separation in pulse periods (P3) for the twomodes, which are (15+2

−6 deg, 6.9 ± 0.1P ) and (23+2−14 deg,

2.9 ± 0.1P ), respectively. Here P denotes the rotational pe-riod of the pulsar. Both modes show power over a broadrange of P3, meaning that the intensity variation does notfollow one constant periodicity. The discovery of two driftingmodes together with their overlap in pulse phase, suggeststhat either the two modes appear at the same time at thesame rotational phase, or that the modes alternate. Follow-ing the description in Serylak et al. (2009), a potential modeswitch corresponding to the two periodicities was searchedfor by calculating the time-resolved 2DFS with a resolutionfrom 10 to 50 pulses. The brightest part of the observationwas examined, but our sensitivity was insufficient to confirmif any transitions occurred on these short timescales.

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6 K. Liu et al.

Figure 7. Statistical results from analysing the pulse stack fromthe entire 15-min observation. The top panel shows the integratedprofile (solid line), longitude-resolved modulation index (pointswith error bars), and longitude-resolved standard deviation (opencircles). The LRFS is plotted below this panel. Below the LRFS,the 2DFS is shown and the power in the 2DFS is vertically in-tegrated and then normalised with respect to the peak value toproduce the bottom plot. Both the LRFS and 2DFS are hori-zontally integrated and then normalised with respect to the peakvalue to produce the left side-panels of the spectra. The rightside-panels label the colour scale (in arbitrary unit) in the LRFSand 2DFS, respectively.

Figure 8. The same LRFS as in Fig. 7, but with the mean powersubtracted in each column. The spectrum is horizontally inte-grated and then normalised with respect to the peak value, pro-ducing the side panel. The right side-panel label the colour scalein the LRFS with power in arbitrary unit.

Previous high-time-resolution observations have shownthat periodic micro-structure in pulse emission is commonfor canonical pulsars (Cordes 1979). The duration of eachmicropulse can range from several hundreds down to sev-eral microseconds (Craft et al. 1968; Bartel & Sieber 1978).However, no such phenomenon has thus far been found inMSPs (e.g. Jenet et al. 1998). To search for periodic micro-structure in PSR J1713+0747, following the definition inLange et al. (1998), we calculated the averaged autocorre-lation function (ACF) over all single pulses compared withthe ACF of the averaged profile in Fig. 9. The highest timeresolution of 140 ns in the data was retained to optimisethe sensitivity to narrow features. Here, because the zero-lag bin of the averaged ACF from single pulses has a powerexceeding 20 times of the other bins, we used the powerin the first non-zero-lag bin to normalise the ACF in bothcases. The comparison between the ACFs suggests that thesingle pulses are significantly narrower than the averagedprofile. The slope of the averaged ACF over all single pulseschanges at around 280 ns (the second non-zero-lag bin). Sim-ilar behaviour in the same type of ACF has already beenwitnessed in canonical pulsars (Hankins 1972; Lange et al.1998). However, here we cannot rule out the cause by instru-mental effect as the change of slope was also seen in the ACFcalculated based on the off-pulse region. No reappearing lo-cal maximum has been seen from either ACF, ruling out thepresence of periodic micro-structure. We further examinedthe ACFs of the brightest 100 single pulses, based on bothpeak flux density and total flux density, and again foundno evidence of micro-structure. In order to rule out poten-tial DM smearing that may lead to the non-detection, werefolded the brightest 10 pulses with different DMs rangingfrom 15.9879 to 15.9983 cm−3pc6, in increments of 4× 10−4

which corresponds to 120 ns smearing time for our observing

6 The central value of the range is 15.9935 cm−3pc, the DM onthe observation epoch derived from Desvignes et al. (2016). Thechosen range also covers the default DM value used in dedispers-ing the data.

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Single pulses of PSR J1713+0747 7

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Figure 9. Averaged ACF of all single pulses (solid line) comparedwith the ACF of the averaged profile (dashed line). The innerpanel shows a zoomed version of the plot in the 0 − 2µs range.Here we retained the highest time resolution of 140 ns for theprofiles.

frequency and bandwidth. The ACFs again showed no signof micro-structure.

3.2 Single pulse polarisation

The polarisation properties of single pulses from MSPs havenot been widely investigated by previous studies due toboth the lack of sensitivity and hardware constraints. Withthe coherently added voltage data as recorded by LEAP, itwas possible to retain full polarisation information for allsingle pulses with proper calibration. Example polarisationprofiles can be found in Fig. 10, where we show the ninehighest-S/N pulses. We found that all these pulses showsignificant linear polarisation, whose fraction with respectto the total intensity appears to be higher when comparedwith the integrated profile (Fig. 2). This indicates that thepolarisation fraction may depend on the brightness of thepulses. For further investigation, in Fig. 11 we grouped thesingle pulses based on their peak flux density and calcu-lated the polarisation fractions from their averages. The biasin linear and circular polarisation was corrected followingSimmons & Stewart (1985) and Tiburzi et al. (2013). Therise of polarisation fraction for both L and V with increas-ing peak flux density is clearly shown. Note that similardependence has already been seen in a handful of canonicalpulsars and another MSP (Mitra et al. 2009; Os lowski et al.2014), while an opposite dependence was also noticed in onecase (Xilouris et al. 1994).

The P.A. curve of the integrated profile in Fig. 2 showsa few discontinuities of approximately 90 deg difference,which usually indicates the existence of orthogonal polar-isation modes (OPMs). Following Gil & Lyne (1995) andOs lowski et al. (2014), in Fig. 12 we calculated the proba-bility density distribution of P.A. measured from single pulsedata, and compared with the P.A. swing obtained from theintegrated profile (Fig. 2). Here we only used samples witha corresponding linear intensity exceeding 3-σ of detection.For a given phase, we formed the number density distribu-tion with each count weighted by the inverse of its measure-ment variance, and normalised the distribution by its sum.

0.3

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Figure 11. Polarisation fractions of linear (top) and circular(bottom) component, as a function of relative peak flux densityof single pulses.

Phase

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-1

Figure 12. Longitude-resolved probability density distributionof P.A. values from single pulses in log-scale. The values wereselected if the corresponding linear polarisation exceeds 3-σ. Thecrosses represent the maximum of probability for each phase bin.The solid line shows the P.A. curve from the integrated profile.Note that between phase 0.59 and 0.60 there is no available P.Ameasurement due to the lack of sufficient linear intensity. All P.A.values were plotted twice (with 360 deg separation) for clarity.

The P.A. values of the single pulses between phase 0.494 and0.530 (mode 2) were found to be rotated by 90 deg comparedto adjacent phase ranges (mode 1), which corresponds to thetwo jumps in the P.A. curve of the integrated profile. Boththe P.A. swing in the integrated profile and the P.A. distri-bution of the single pulses show a rapid change around phase0.6. However, the low degree of linear polarisation makes itimpossible to distinguish between a smooth transition or adiscontinuity. In either case the P.A. changes by less than90 deg.

Within some ranges of pulse phase, in particular thosecorresponding to mode 1, the most-likely value of P.A. iswidely scattered, covering a range of nearly 90 deg. For sev-

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8 K. Liu et al.

0

0.2

0.4

0

0.2

0.4

0

0.2

0.4

0.49 0.50 0.51 0.49 0.50 0.51 0.49 0.50 0.51

Phase

Inte

nsity

(ar

b. u

nit)

Figure 10. Polarisation profiles of the nine highest-S/N pulses. The definition of the lines is the same as in Fig. 2. The time resolutionin this plot is 2.3µs.

eral selected phase bins, we checked the number densitydistributions of P.A. which turned out to be spread overat least a few tens of degrees, broader than what is ex-pected purely from instrumental noise (less than ten de-grees). The widths of the distributions grow as the linearintensity decreases. Such phenomena have already been no-ticed in many canonical pulsars (e.g., Stinebring et al. 1984;McKinnon & Stinebring 1998; Karastergiou et al. 2002). Toexamine if the distribution of P.A. are different in the twomodes, we chose pulse phases of the same linear intensity inthe integrated profile, and plotted the distribution of P.A.values with measurement error less than 7 deg7. The resultis shown in Fig. 13, where the distribution from mode 1is seen to be broader than in mode 2. This also indicatesthe existence of an intrinsic P.A. distribution instead of asingle value. A recovery of the intrinsic distribution, how-ever, would require a detailed modelling of the instrumentalnoise which additionally takes into account the covarianceof the noise component in the measured Stokes Q and U

(van Straten 2010).

3.3 Impact on timing

The variability of single pulses introduces a variation in theintegrated profile. This manifests itself as stochastic fluctu-ations in the derived pulse time-of-arrivals (TOAs), which iscommonly known as phase jitter. Shannon & Cordes (2012)have shown that the timing precision of PSR J1713+0747on short timescales can be greatly limited by phase jitter.The subsequent jitter noise in the timing data scales as thesquare-root of number of averaged pulses, as expected from

7 The measurement uncertainties were obtained by standard er-ror propagation using the variance of Stokes Q and U . It is ex-pected to give the 1-σ error, when the S/N of the linear intensityexceeds 3 which corresponds to an error of approximately 9.6 degin P.A. (Wardle & Kronberg 1974).

0

4

8

12

16

20

-80 -60 -40 -20 0 20 40 60 80

P.A. (deg)

Per

cent

Mode 1Mode 2

Figure 13. Number densities of P.A. (normalised by the sum)from pulse phase 0.475 (dashed line) and 0.522 (solid line) whichcorrespond to different orthogonal polarisation modes. The linearintensities in the integrated profile are the same at the two phases.

the theoretical model (e.g. Cordes & Downs 1985). Follow-ing the method described in Liu et al. (2012), we measuredjitter noise of 494 ns in our data for a 10-s integration time,in agreement with the finding in Shannon & Cordes (2012).Consistent results were also found when we made the mea-surements with data from each 16-MHz sub-band instead ofusing the full bandwidth.

The phase jitter of integrated profiles is usually assumedto follow a Gaussian distribution. However, if single pulsesare not normally distributed in rotational phase, the result-ing jitter noise is likely to deviate from Gaussian noise, espe-cially when the number of averaged pulses is not sufficientlylarge. For PSR J1713+0747, the asymmetric occurrence rateof single pulses in rotational phase is clear from Fig. 5, which

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Single pulses of PSR J1713+0747 9

10-4

10-3

10-2

10-1

100

101 102 103

Number of pulses

P-v

alue

All

p1

p2

p3

Figure 14. Measured p-values from standard K-S test ofweighted timing residuals based on different numbers of averagedpulses, obtained from all pulses as well as sub-sets of pulses basedon their peak flux densities. Details of the three sub-sets can befound in Table. 2.

was also shown in Shannon & Cordes (2012). To further in-vestigate the nature of the jitter noise, we produced tim-ing residuals based on different numbers of averaged pulses,weighted them by the sum of radiometer and jitter noise,and calculated their p-values8 from a standard K-S test un-der the hypothesis of a normal distribution (Fig. 14). We cansee that the phase jitter starts to appear as Gaussian noiseonly after integrating ∼ 102 pulses. Integrating a sub-set ofpulses selected based on a chosen range of peak flux den-sities gives a relatively quick conversion, as expected. Thisplaces a lower limit on the number of pulses (i.e., ∼ 102)to be integrated in timing analysis to avoid significant non-Gaussianity. Note that a similar limit was already derivedfor PSR J0437−4715, based on simulations of the regularityof profile shapes (Liu et al. 2011).

Single pulses do not necessarily have the same variabil-ity as a function of rotational phase (e.g., Cognard et al.1996; Liu et al. 2015), meaning that their contribution tothe resulting jitter noise can be different. In Os lowski et al.(2014), single pulses from PSR J0437−4715 were groupedbased on their peak flux density and averaged into inte-grated profiles, which did not result in a significant improve-ment of the timing precision. Following this, we divided thepulses into groups based on their relative peak flux densities,relative total flux densities and widths, separately. Subse-quently, we produced timing residuals containing the samenumber of TOAs from each group9. The results are summa-rized in Table 2. It can be seen that all sub-sets contributeapproximately the same jitter noise per pulse, though pulses

8 Such a p-value close to unity shows that we cannot detect sig-nificant deviation from the normal distribution of data.9 The TOAs were estimated with the standard template match-ing algorithm (Taylor 1992). To time each sub-set of pulses, weformed the template by integrating the whole sub-set and per-formed wavelet-smoothing to the integrated profile, to avoid self-correlation by the noise. The residuals were calculated with theephemeris from Desvignes et al. (2016), without any fit for thetiming parameters.

Table 2. Timing residuals achieved with different sub-sets ofpulses. The pulses are grouped with respect to their relative peakflux density (p, peak flux density divided by its running mean),relative flux density (f , total flux density divided by its runningmean) and pulse width (w, defined as W50). The rms values arecalculated based on the same number of TOAs.

Range Percent RMS (µs) χ2re σj,0 (µs)

p1 (0, 1.17) 35.4 1.01 2.75 22.5

p2 (1.17, 1.60) 31.3 1.02 8.20 25.1p3 (1.60, 3.00) 29.3 0.816 18.9 20.2p4 (3.00,+∞) 3.93 1.97 69.4 18.3

f1 (0, 0.77) 31.7 1.04 2.87 22.1f2 (0.77, 1.05) 26.6 1.11 9.98 25.4f3 (1.05, 1.39) 25.7 0.862 10.5 19.5f4 (1.39,+∞) 16.0 0.935 18.5 17.1

w1 (0, 30)µs 33.9 1.06 3.70 21.1w2 (30, 50)µs 35.5 0.833 11.3 21.2w3 (50,+∞)µs 30.6 0.867 17.8 21.2

All (0,+∞) 100 0.590 12.9 26.5

of high flux density and narrow width tend to show slightlylower values. The achieved timing rms values are also sig-nificantly higher than when using all pulses. Thus, there isno apparent benefit in grouping the single pulses for thepurpose of precision timing.

In principle, the achievable TOA precision should beindependent of the choice of time resolution (δτ ) in form-ing the integrated profile, as far as its features are fullyresolved (e.g., Downs & Reichley 1983). Here we used thesingle pulse data to form integrated profiles of 1-s integra-tions and variable time resolution, and calculated their cor-responding TOA uncertainties. On average, the TOA un-certainty drops no more than 5% when δτ decreases from8.9µs to 140 ns. This suggests that the narrow features inthe single pulses are mostly averaged out, and a time reso-lution beyond 10µs is sufficient in shaping the profiles whenthe exposure time is longer than one second.

4 CONCLUSIONS AND DISCUSSIONS

We have studied the properties of single pulses fromPSR J1713+0747 using 15 minutes of data (corresponding to∼ 197, 000 pulses) collected by LEAP when the pulsar wasextremely bright. The pulse energy distribution is shownto be consistent with a log-normal distribution. The pulsewidths are typically around 0.04 ms for the bright pulses.In addition, we have confirmed the detection of periodicintensity modulation by Edwards & Stappers (2003), andin addition revealed its association with systematic driftingsub-pulses. From the 2DFS, P2 and P3 of the two modeswere measured to be (15+2

−6 deg, 6.9± 0.1 P ) and (23+2−14 deg,

2.9 ± 0.1P ), respectively. The occurrence of the two modesis found to overlap at the phase of the main peak in theintegrated profile, providing the first evidence for super-posed modes of drifting sub-pulse in MSPs. The mode at0.14 cpp was also apparent at the trailing edge of the inte-grated profile, where the other was not detected. We did notfind any evidence of periodic pulse micro-structure with a

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10 K. Liu et al.

time resolution of up to 140 ns. With full polarisation data,we have shown that the bright pulses are significantly lin-early polarised. On average, the fractional polarisation of thepulses increases with increasing pulse peak flux density. Us-ing single pulse polarisation, we have presented a longitude-resolved probability density map of P.A., and shown theexistence of two orthogonal modes of polarisation that areclearly distinct in pulse phase. Finally, we found that jitternoise induced by pulse variability can be described as Gaus-sian only when at least 100 pulses have been integrated. Sta-tistically, pulses of different properties contribute equally tothe resulting jitter noise, though pulses with high peak fluxdensity have slightly less effect. By timing sub-sets of pulseswith respect to their relative peak flux density, relative totalflux density and pulse width, we did not find any significantimprovement on the overall timing precision. Changing thetime resolution used on 1-s integrations from 8.9µs to 140 nsdoes not result in significant difference in their TOA un-certainties, meaning that most features on those integratedprofiles are resolved with time resolution beyond 10µs.

Micro-structure in pulsed radio emission has mostlybeen searched for in canonical pulsars, and most of thesearches that have sufficient time resolution and sensitiv-ity have achieved successful detection (e.g. Lange et al.1998; Mitra et al. 2015). On the other hand, besidesPSR J1713+0747, micro-structure has also been searched forin another two MSPs, PSR J0437−4715 and PSR B1937+21,with no detection (Jenet et al. 1998; Jenet et al. 2001). Us-ing observations of canonical pulsars, Kramer et al. (2002)established a relation between micro-structure width andpulsar rotational period. A simple extrapolation from thatrelation leads to a micro-structure width of 0.5 − 10µs forMSPs, well above the time resolutions of the data used forthe searches in MSPs. Thus, the non-detections may implythat either micro-structure is less common in MSPs com-pared with the situation in canonical pulsars, or the rela-tion between micro-structure width and rotational periodis different in MSPs. Nevertheless, this needs to be furtherverified by more sample studies.

The discovery of drifting sub-pulses in PSR J1713+0747increases the number of MSP “drifters” to three,together with PSR J1012+5307 and J1518+4904(Edwards & Stappers 2003). As high-sensitivity searchesfor drifting sub-pulses have been performed in no more thanten MSPs, drifting sub-pulses do not seem to be an unusualphenomenon among this part of the pulsar population. Itis interesting to note that all three exhibit a quasi-periodicdrifting fashion, and are classified as “diffuse” drifter by thedefinition in Weltevrede et al. (2006). Given that nearly halfof the drifters in canonical pulsars show a precise driftingperiod (referred to as “coherent” drifters), quasi-periodicdrifting may be comparatively more common in MSPs.Again, robust statistics of the ratio requires future workincluding an expanded number of samples. Searching fordrifting sub-pulses in more MSPs will also show if thephenomenon is correlated with any pulsar properties, suchas the characteristic age which has been discovered amongthe population of non-recycled pulsars (Weltevrede et al.2006, 2007).

Despite the vast increase in sensitivity delivered byLEAP, a thorough understanding of single pulse properties,in particular those from the lower end of the energy distri-

bution, is still limited by S/N. The situation is likely to besignificantly improved when coherent addition of the pulsesignal is achieved with more telescopes, as also suggestedin Dolch et al. (2014). Eventually, the next generation ofradio telescopes, e.g., the Square Kilometre Array and theFive hundred meter Aperture Spherical Telescope, will pro-vide the best opportunity ever to study single pulse emissionfrom MSPs.

The behaviour of pulse variability is seen to be a source-dependent phenomenon among pulsars including MSPs. De-spite the lack of improvement in the timing precision ofPSR J1713+0747, single pulse data could still be used tomitigate jitter noise in some other pulsars, especially whenpulses with different properties exhibit different distribu-tions in phase. In this case, a weighting scheme concerningtheir contribution to jitter noise may be introduced whenintegrating the pulses, so as to optimally use the signalof the pulses which show less variability in phase. An ap-proach in a similar framework has already been discussed inImgrund et al. (2015). Nevertheless, implementation of suchmethods is yet to be thoroughly investigated.

ACKNOWLEDGEMENTS

We would like to thank S. Os lowski, A. Noutsos andJ. M. Cordes for valuable discussions, I. Cognard for assis-tance in observations at Nancay, and A. Jessner for care-fully reading the manuscript and providing valuable in-sights. We also thank the anonymous referee for provid-ing constructive suggestions to improve the article. K. Liuacknowledges the financial support by the European Re-search Council for the ERC Synergy Grant BlackHoleCamunder contract no. 610058. This work is supported by theERC Advanced Grant “LEAP”, Grant Agreement Num-ber 227947 (PI M. Kramer). The Effelsberg 100-m tele-scope is operated by the Max-Planck-Institut fur Radioas-tronomie. The Westerbork Synthesis Radio Telescope is op-erated by the Netherlands Foundation for Radio Astronomy,ASTRON, with support from NWO. The Nancay Radio Ob-servatory is operated by the Paris Observatory, associatedwith the French Centre National de la Recherche Scien-tifique. CGB and SS acknowledges support from the Eu-ropean Research Council under the European Union’s Sev-enth Framework Programme (FP/2007-2013) / ERC GrantAgreement nr. 337062 (DRAGNET; PI Jason Hessels). RSacknowledges the financial support from the European Re-search Council under the European Union’s Seventh Frame-work Programme (FP/2007-2013) / ERC Grant Agreementn. 617199. KJL gratefully acknowledges support from Na-tional Basic Research Program of China, 973 Program,2015CB857101 and NSFC U15311243. This research is a re-sult of the common effect to directly detect gravitationalwaves using pulsar timing, known as the European PulsarTiming Array (EPTA).

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